366 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATION'S 



ft'! = f COS %, 



x 2 = T sill a! COS a», 



x 3 = r sin aj sin a> cos 03, 



Zj — r sin aj sin a, . . . . sin Oi_i cos 04 

 :r n = /• sin aj sin 02 . . . . sin <*„_!. 



From these we have 



r = (z? + xi + .... + x*)\ 



a", 

 COS ai = - . 

 1 r 



a-., 



(14) 



' (15) 



1 1 



Uma n _ 1 = ~, 



•*» — 1 .... 



Now we have at once (writing for the moment oq = r) 



+ . 



(7<i , d<f> . . d<f> 



+ d^ V%> + ^ V"«i + ^ \ro3 + . . . . (16) 



The summations in this formula are for all values of J: from 1 to i. By performing 

 the necessary operations we find quite readily 



~ ^7a-,J — a^n sin 2 a,' 3 — \> % * — \ 



